Attaining the optimal constant for higher-order Sobolev inequalities on manifolds via asymptotic analysis

TL;DR

This paper establishes the sharp constant in higher-order Sobolev inequalities on manifolds, using asymptotic analysis of solutions to critical polyharmonic equations to understand the inequality's behavior.

Contribution

It provides the first proof of the optimal Sobolev constant on manifolds via asymptotic blow-up analysis of polyharmonic equations.

Findings

Existence of a sharp Sobolev constant on manifolds.

Explicit asymptotic behavior of solutions as parameter tends to infinity.

Pointwise description of solutions to critical polyharmonic equations.

Abstract

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2} u|^2 \,dv_g + B_0 \|u\|_{H^{k-1}(M)}^2,\] where $2^\sharp = \frac{2n}{n-2k}$ and $\Delta_g = -\operatorname{div}_g(\nabla\cdot)$. Here $K_0$ is the optimal constant for the Euclidean Sobolev inequality $\big(\int_{\mathbb{R}^n} |u|^{2^\sharp}\big)^{2/2^\sharp} \leq K_0^2 \int_{\mathbb{R}^n} |\nabla^k u|^2$ for all $u \in C_c^\infty(\mathbb{R}^n)$. This result is proved as a consequence of the pointwise blow-up analysis for a sequence of positive solutions $(u_\alpha)_\alpha$ to polyharmonic critical non-linear equations of the form $(\Delta_g + \alpha)^k u = u^{2^\sharp-1}$ in $M$. We obtain a pointwise description of $u_\alpha$, with explicit dependence in $\alpha$ as $\alpha\to \infty$.